﻿in Small Orbits. ' 513 



Now, from the principle of equilibrium in fluid masses, 

 Xdx+Ydy+Zdz=iO. 



Substituting the above values for X, Y, and Z, and integrating, 

 there results 



N mk* 3 W W 



1N " (* + ,«+*)* "2B^ + W -°' * * (5) 



the equation of the surface of the satellite N being the arbitrary- 

 constant required in integration. 



To find the relation between the principal axes, it is necessary 

 to deduce from the last equation an expression for the maximum 

 value of each ordinate, the other two being successively made 

 equal to 0. We thus obtain 



mk* 3M* 2 A 2 M _ 



-A- + -W- = N ' (6 > 



^ = N, (7) 



mk* MFC 2 

 C 2D 3 "" w 



Subtracting (6) and (8) successively from (7), we find 



rf(A-B) 3MFA 2 . ,, 3MA 8 B 



ict> ~= rt m j or A— B= n TV , , . (9) 



AB 2D 3 2mD 3 ' v ' 



mF(B-C) MFC 2 _ n MC 3 B 



BC = -m*-> 0i ' B - C =2mW'> ' ' < 10 > 



and adding (9) and (10), 



"=5^ (") 



From a comparison of the last three equations, it appears that 

 when the extraneous force and the consequent deviation from a 

 sphere is small, the ellipticities of the principal sections of the 

 satellite would be nearly in the ratio of four, three, and one. 



The greatest degree in which polar compression could be ex- 

 hibited in such cases would evidently be reached when gravity 

 was entirely annulled at the part of the surface in conjunction 

 with the primary, so that 



mk 9 3MFA 

 A 2 D 3 



To find, therefore, the greatest disproportion which can occur 



u i a ^w^u U , 4J „ 3MFA 2 , mk 9 



between A and x>, let -^-r- be substituted tor — ~~ru — , and -=r - 



